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Friday, June 01, 2007

Bouncing Neutrons in the Gravitational Field

I remember a moment of excitement and puzzlement early on in my first class in quantum mechanics, when our professor announced that now, he would discuss the "freier Fall" in quantum mechanics. I was excited, because it seemed great to me to transfer such an elementary situation as the free fall of a stone into the realm of quantum mechanics, and puzzled, because I knew that the gravitational potential is so extremely weak that it can be safely ignored on scales where quantum mechanics comes into play - at least, in most cases. Alas, that lecture was quite a disappointment, because of an ambiguity of the German wording "freier Fall": it can mean both the free "fall", and the free "case" (as in Wittgenstein's famous dictum "Die Welt ist alles, was der Fall ist.") - and what we learned in our lecture was just about the free case, the quite boring plane wave motion of a quantum particle subject to no potential whatsoever.

What we did not learn in our class was that, even back at that time, there had been several clever experiments with neutrons which demonstrate the influence of the gravitational potential on the phase of the neutron wave function using interferometers. Neutrons, of course, are ideal particles to perform such experiments, since they have no electric charge and are not subject to the influence of the ubiquitous electromagnetic fields.

But only over the last few years, new experiments have been realised that show directly the quantisation of the vertical "free fall" motion of neutrons in the gravitational field of the Earth. I had heard about them some time ago in connection with their possible role for the detection of Non-Newtonian forces, or the modifications of Newtonian gravity at short distances. Then, earlier this year, I heard a talk by one of the experimenters at Frankfurt University, and I was quote fascinated when I followed the papers describing the experiments.

The essential point of these experiments is the following: If you prepare a beam of very slow neutrons - with velocities about 10 m/s - you can make them hop over a reflecting plane much like you can let hop a pebble over the surface of a lake. Then, you can observe that the vertical part of the motion of the neutrons - with velocities smaller than 5 cm/s - is quantised. In fact, one can detect the quantum states of neutrons in the gravitational field of the Earth! Let me explain in more detail...

Free Fall in Classical Mechanics ...

In order to better understand the experiment, let's go back one step and consider the very simple motion of an elastic ball which is dropped on the ground. If the ground is plane and reflecting, and the ball is ideally elastic such that there is no dissipation of energy, the ball will jump back to the height of where it was dropped from, fall down again, jump back, fall, and so on. The height of the ball over ground as a function of time is shown as the blue curve in the left of this figure: it is simply a sequence of nice parabolas.

We can now ask, What is the probability to find the bouncing ball in a certain height above the floor? For example, we could make a movie of the bouncing ball, take a still at some random time, and check the distribution of the height of the ball if we repeat this for many random stills. The result of this random sampling of the bouncing motion of the ball is the probability distribution shown in red on the right-hand side of figure. The probability to find the ball at a certain height in this idealised, "stationary" situation, where the elastic ball is bouncing forever, is highest at the upper turning point of the motion, and lowest at the bottom, where the ball is reflected.

... and in Quantum Mechanics

So much for classical mechanics, as we know it from every-day life. In quantum mechanics, unfortunately, there is not anymore such a thing as the path of a particle, with position and velocity as well-defined quantities at any instant in time. However, it still makes sense to speak of stationary states, and of the probability distribution to find a particle at a certain position. In quantum mechanics, it is the wave function which provides us with this probability distribution by calculating its square. And the law of nature determining the wave function is encoded in the famous Schrödinger equation. The Schrödinger equation for a stationary state is an "eigenvalue equation", whose solution yields, at the same time, the wave function and the value of the energy of the corresponding state. For the motion of a particle in a linear potential - such as the potential energy mgx of a particle with mass m at height x above ground in the gravitational field with acceleration g at the surface of the Earth - it reads

In some cases, there are so-called "exact solutions" to the Schrödinger equation - wave functions that are given by certain functions one can look up in thick compendia, or at MathWorld. These functions usually are some beautiful beasts out of the zoo of so-called "special functions". Such is the case for the motion of a particle in a linear potential, where the solution of the Schrödinger equation is given by the Airy function Ai(x). Interestingly, this function first showed up in physics when the British astronomer George Airy applied the wave theory of light to the phenomenon of the rainbow...

Quantum States of Particles in the Gravitational Field

As a result of solving the Schrödinger equation, there is a stationary state with a minimal energy - the ground state - and a series of excited states with higher energies. Here is how the wave function of the second excited state of a particle in the gravitational field looks like as a function of the height above ground:

The wave function, shown on the left in magenta, oscillates through two nodes, and goes down to zero exponentially above the classical limiting height, which corresponds to the upper turning point of the parabola of a classical particle with the same energy. For neutrons in this state, this height is 32.4 µm above the plane. The green curve on the right shows the probability density corresponding to the wave function. It is quite different from the classical probability density, shown in red. As a characteristical property of a quantum system, there is, besides the two nodes, a certain probability to find the particle above the classical turning point. This is an example of the tunnel effect: there is a chance to find a quantum particle in regions where by the laws of classical physics, it would not be allowed to be because of unsufficient energy.

However, going from the ground state to ever higher excited states eventually reproduces the probability distribtion of classical physics. This is what is called the correspondence principle, and you can see what it means if you have a look at the wave function for 60th excited state: Here, the probability distribution derived from the quantum wave function follows already very closely the classical distribution.

So far, we have been talking about theory: the Schrödinger equation and its solutions in guise of the Airy function. There is no reason at all to doubt the validity of the Schrödinger equation: it has been thoroughly tested in innumerable situations, from the hydrogen atom to solid state physics. However, in all these situations, the interaction of the particles involved is electromagnetic, and not by gravitation. For this reason, it is extremely interesting to think about ways to check the solution of the Schrödinger equation for particles in the gravitational field. As we have seen before, the best way to do this is to work with neutrons, in order to avoid spurious electromagnetic effects.

Bouncing Neutrons in the Gravitational Field

Unfortunately, it is so far not possible to scan directly the probability distribution of neutrons in the gravitational field. However, in a clever experimental setup, one can look at the transmission of neutrons through a channel between with a horizontal reflecting surface where they can bounce like pebbles over a lake, and an absorber ahead. This is a rough sketch of the setup:

The decisive idea of the experiment is to vary the height of the absorber above the reflecting plane, and to monitor the transmission of neutrons as a function of this height. If the height of the absorber is to low, the ground state for the vertical motion of the neutrons does not fit into the channel, and no neutrons will pass the channel. Transmission sets in once the height of the channel is sufficiently large to accommodate the ground state wave function of the vertical motion of the neutron. Moreover, whenever with increasing height of the channel, one more of the excited wave function fits in, the transmission should increase. The first of these steps, and the corresponding wave functions and probability densities, are shown in this figure:

The interesting point now is, can this stepwise increase of transmission be observed in actual experimental data? Here are measured data, and indeed - the first step is clearly visible, and the second and third step can be identified:

This has been the first verification of quantised states of particles in the gravitational field!

What can be learned

You may wonder if the experiment may not have shown just some "particle in a box" quantisation, since the channel for the neutrons formed by the reflecting plane and the absorber may make up such a box. This objection has been raised, indeed, in a comment paper, and has been answered by detailed calculations, and improved experiments: the conclusion about quantisation in the gravitational field remains fully valid!

However, limits about modifications of Newtonian gravity from this experiment remain restricted. Such a modification would change the potential the neutrons are moving in. For example, a short-range force caused by the matter of reflecting plane could contribute to the potential of the neutrons. However, as comes out, such an additional potential would be very weak and have nearly no influence at all on the overall wave function of the neutron.

Moreover, it is clear that in this experiment, the gravitational field is always a classical background field, which itself is not quantised at all. There may be the possibility that a neutron undergoes a transition from, say, the second to the first quantised state, thereby emitting a graviton - similar to the electron in an atom, which emits a photon when the electron makes a transition. Unfortunately, this probability is so low that it is not reasonable to expect that it may ever be measured....

But all these restrictions do not change at all the main point that this a very exciting, elementary experiment, which could find its way into textbooks of quantum mechanics!

Quantum motion of a neutron in a wave-guide in the gravitational field by A.Yu. Voronin, H. Abele, S. Baessler, V.V. Nesvizhevsky, A.K. Petukhov, K.V. Protasov, A. Westphal; Phys.Rev. D73 (2006) 044029 (doi: 10.1103/PhysRevD.73.044029 | arXiv: quant-ph/0512129v2) - A long and detailed discussion of point such as the "particle in the box" ambiguity and the role of the absorber.

Constrains on non-Newtonian gravity from the experiment on neutron quantum states in the Earth's gravitational field by V.V. Nesvizhevsky, K.V. Protasov; Class.Quant.Grav.21 (2004) 4557-4566 (doi: 10.1088/0264-9381/21/19/005 | arXiv: hep-ph/0401179v1) - As the title says: a discussion of the constraints for Non-Newtonian forces.

Spontaneous emission of graviton by a quantum bouncer by G. Pignol, K.V. Protasov, V.V. Nesvizhevsky; Class.Quant.Grav.24 (2007) 2439-2441 (doi: 10.1088/0264-9381/24/9/N02 | arXiv: quant-ph/07702256v1) - As the title suggests: the estimate for the emission of a graviton from the neutron in the gravitational field.

40 comments:

Wow! Thanks so much for sharing that---that's a pretty exciting result!

It's the best evidence for quantum gravity actually being a truly "quantized gravity" theory that I've ever seen. It's also fun to keep seeing MOND and co. pop up and say "hey, we might be on to something!" I wonder what role MOND will have in our eventual theory.

This is a wonderful write up! You should get an award. Lot's of work, and it's beautiful.

"Moreover, it is clear that in this experiment, the gravitational field is always a classical background field, which itself is not quantised at all."

When I turn on my buddy's TV and receive signals for Gilligan's Island the reception is purely classical, but photons are nevertheless quantized. It's a matter of the coarseness of the experiment.

"There may be the possibility that a neutron undergoes a transition from, say, the second to the first quantised state, thereby emitting a graviton - similar to the electron in an atom, which emits a photon when the electron makes a transition. Unfortunately, this probability is so low that it is not reasonable to expect that it may ever be measured."

If one imagines that gravitons are the cause of gravitational attraction (on a flat space, perhaps), then it is clear that in this experiment, there must be such large numbers of gravitons emitted and absorbed by the neutrons that their wave functions are constantly modified. Otherwise one couldn't model the gravitational field as a simple term in Schroedinger's equation. But that's not what Pignol et al calculated. What they calculated was the probability of a huge quantum jump in state, due to graviton emission. This is like calculating the probability of getting the Dick van Dyke Show on the Gilligan's Island channel.

Let me put my point another way. You know that an electron is bound to a proton by exchange of huge numbers of virtual photons. The events where an on mass shell photon is emitted by a hydrogen atom changing state are, by comparison, quite rare. Just because the transition is forbidden (and rare) doesn't mean that the virtual partciles are not there, or that the electric force should be modeled as an effect of pure geometry (that is, not quantized).

thanks :-) I received the printed version, but hadn't seen the online version. Ha, I should send that to my landlord.

Hi Stefan:

I have only now had the time to fully read the text. This is a really excellent explanation! You know that I read their papers, as well as some other articles on the experiment, but none of them was a clearly as yours :-)

Hi Domenic:

This has nothing to do with quantizing gravity. It's a quantum effect in a classical field. I have my own opinion about MOND (not very positive), but currently too tired to elaborate on it. Maybe some other post...

aah, I just realized 'anonymous' above said almost exactly what I said (sorry folks, I am somewhat confuzzled these days). Anyway, there is something one can learn from that, it tells us that qm in the grav field works exactly how predicted. okay, that's somewhat boring, but consider that most of the time we observe quantization in an electric field (energy levels of atoms) not a gravitational field. I recall when I heard about the experiment with the bouncing neutrons the first time, they made a claim about testing Newton's law (was around the time when these tests were en vogue because of higher-dim. theories potentially causing deviations from it), but I thought this is more an attempt to sell the topic than a really good experiment to check Newton's law (Stefan and I, we were talking about that yesterday and we agreed that the sensitivity to the grav. attraction of the lower reflector is too small to have a significant influence on the measured quantities). Best,

excuse that I am a slow at understanding physics. But if I understoof everything, correct. We have states of vertical motion $E_0, E_1, ...$. To these energies one links using the Hamiltonian involving gravity heights. If these heights stay low enough the neutrons are not absorbed. If they are too big, they are absorbed.

The next thing, I wonder about is why are you allowed to choose the decaying solution? Since the differential equation would have two solutions, of which one is growing?And on one more side now, how do you implement the mirror into the differential equation? Some kind of boundary condition?

Lol Bee the heating in your apartment still in the ceilingBoiling a kettle on the Moon for a cup of tea should be fun.PS - What are the most obvious advantages and disadvantages of living in low gravity environs?Other than being able to jump higher & further - cold feet?

What if the "movable neutron absorber" was replaced with a devise such as a "cone shaped neutron absorber", dissected in half, and placed end_to_end so a s to act like to funnels?..then from one side (left) a Proton is sent in, and at the same time from the (right) a Neutron is sent in?

Due to the difference in size of the Proton and Neutron, they should go through the central core aparture, without colliding?Whereas two identical particles would have a high probability of colliding?

The "neutron absorber plate", is the wrong shape, should it not be half-cone ? the probability density of the outgoing neutrons should be compared for a "flat" and "curved" absorbtion plate?

Hi Domenic:you're more than welcome. Actually, I am very happy about your comment because I told Stefan repeatedly, he should make really really clear it has nothing to do with quantizing gravity, even though there is gravity and quantization involved. And sure enough the first comment gave me this nice feeling of I-told-you-so ;-) Best,

Since the differential equation would have two solutions, of which one is growing?And on one more side now, how do you implement the mirror into the differential equation? Some kind of boundary condition?

It's a wave-function so it should be square integrable (well, at least it should not diverge). Yes, the mirror is a boundary condition. Best,

thanks that you like the post :-)... Bee had kept asking me about it over the last weeks, and while she was busy preparing her talks for the next weeks, I finally put together the write-up... So I am glad that it was worth the effort..

thank you for your detailed discussion of "virtual" versus "real" gravitons in the experiment. I see that my wording the gravitational field is always a classical background field, which itself is not quantised at all could have been more to the point - what I wanted to say, as you guess, is the following:

In this experiment, the gravitational field enters as classical background field. Whether it is a classical field and nothing else, or, in fact, originating from a constant exchange of virtual gravitons or whatsoever does not matter for the experiment - and can not even be discriminated in the experiment.

The experiment shows the quantisation of neutron states in the gravitational potential gh, and that's it.

However, since no one before could verify experimentally that the solutions of the Schrödinger equation with the gravitational potential indeed describe nature, this is an interesting result!

The Pignol at al. paper, I understand its idea in this way: from the observation of quantised neutron states alone, one cannot say anything about the quantisation of the gravitational field. If, however, one could observe a transition of the neutron between two states in the gravitational field, this would be a very strong hint that the gravitational field itself is quantised, since this transition could be understood best by the emission of a real graviton. As they conclude in the paper, "This effect is unfortunately experimentally too rare to observe." (this final phrase is not in the preprint, btw)

hm... there has indeed always been confusion about "quantisation of the gravitational field" versus "quantisation in the gravitational field" in this experiment - I remember that back from the first press releases I had read about it, some of which had been quite misleading...

Anyway, I hope it is now abundantly clear (thank you, anonymous, and Bee :-) that the experiment is about "quantisation in the gravitational field", and that this in itself, while not yet touching the "holy grail" of quantum gravity, is interesting enough.

I am not quite sure what you mean here. For the experiment, even if the electrostatic Coulomb force could be switched off (very difficult to do in practice), small differences in size of the neutron and the proton do not matter at all. The length scale of the Airy ground state wave function is on the order of 10 micrometer, or ten orders of magnitude larger than the diameter of the neutron and proton. The percent effects of the extensions of the wave function from the mass difference between proton and neutron are washed out by mirror- and absorber imperfections.

Whereas two identical particles would have a high probability of colliding?

But neutrons are fermions anyhwo - I am not so sure what this yould mean...

should not there be a "flat" as well as "curved" neutron absorbtion plate, as gravity is both

again, I do not understand what you want to say - but anyhow: on the scale of the experiment (about a Meter), you can completely forget about Einstein and rely on Newton for gravity...

Yes, you've got it right. For each of the states, you know exactly at which height the wave function switches from the oscillatory to the exponentially damped behaviour. This is exactly at the height which would be the upper turning point of a bouncing particle in the gravitational field with energy corresponding to the energy eigenvalue of that state.

These heights are 13.7 micrometer for the ground state, 25 micrometer for the first excited state, 32.4 micrometer for the second excited state, and so on...

Now, if the absorber is at a distance less than about 13.7 micrometer above the reflector, there is a very high chance that all neutrons will be absorbed, because they will hit the absorber, and thus, no transmission.

Once the height is more than the 13.7 micrometer, those neutrons whose motion in the vertical direction is given by the ground state have a high probability not to be absorbed, and transmission through the channel sets in. If the height of the channel is 25 micrometer, the neutron with vertical motion in the ground state and the first excited state can pass through the channel, so transmission raises further, and so on...

Of course, the wave function even of the ground state has some (minute) non-zero value at arbitrary height, and there is always a chance for the neutron to be absorbed. You can just calculate probabilities of transmission through the channel without absorption. As a result, the steps in the transmission are washed out.

Moreover, one should take into accout the influence of the absorber on the wave function. This has been done in detail in quant-ph/0512129.

Which brings me to your question about the boundary condition:

Of course, when solving the stationary Schrödinger equation, you have to specify some boundary conditions. In this case, you are interested in bound states, so the solution should be normalizable. Or, more physically speaking, the probability to find the particle at arbitrary height above the ground should be very very small for a finite energy. Thus, of the Airy functions Ai and Bi, the function Bi is not a physically reasonable solution, since it would mean exponential growth with height of the probability. So much for the "upper end" of the wave function.

Your question why, at the "lower end", the wave function should have a node at the reflector is a good one.

The reason is that a wave function in general should not have jumps (if I remember correctly, that's because you want the probabilty density to obey a continunity equation), and because the wave function is zero in the reflector, it has to have a node at the surface of the reflector. This is similar to the reflection of an elastic wave on a string at a fixed end.

So, you shift the Airy function Ai in such a way that the reflector is at a node, and by this procedure you get all the wave functions for the ground state and the excited states.

When you say that "only the neutron is quantized" I can't disagree, but that's not the usual way one would talk about electrons in a atom. Instead people usually say that the atomic system is quantized. So why not say the same about the Earth-neutron system?

The difference, I guess, is that the electromagnetic modes of the atomic system are commonly enough excited to be observable - we do see the photons.

Sakurai's undergraduate textbook, "Modern Quantum Mechanics", claims that gravity has been experimentally observed in Schroedginer's equation. See pages 123-43, where they discuss potentials and gauge transformations. This is in chapter 2 section 6. A short description of this chapter has been kindly typed onto the web by Patrick Van Esch.

My opinion was that the analysis by Sakurai was wrong. I don't have the text at hand. I recall that the experimenters built a rigid device that showed neutron interference between two beams. As one rotated the device, the two paths had different gravitational potentials and this causes an interference shift. This was in the section on gauge transformations because, in Schroedinger's equation, a change of overall energy changes the wave functions without changing any calculations and therefore is a gauge transform.

My complaint was that if one does not wish to put a gravitational term in Schroedinger's equation, one can calculate the same interference result by taking into account how altitude alters local clocks. Time spent at altitude changes phase. Either way of doing the calculation gives the same result. I suspect that the same criticism could be applied to these experiments too. I.e. solve Schroedinger's equation on mildly curved spacetime with no gravity term and you'll get the same result. Or for a constantly accelerated frame of reference.

but sadly we didn't discuss it any further or talked about an actual experiment.

that's, in my opinion, a very general, sad feature about quantum mechanics classes.

The point is that knowadays, there are cool and real experiments to many of the standard topics in quantum mechanics textbooks - take quantum wires/dots for the particle in the well potential, or experiments with single atoms in traps, or the mapping of molecular orbitals with picosecond lasers... but they are often not mentioned in texts. The time where the only experiments discussed in textbooks are the blackbody spectrum, the photoeffect, the Compton effect, and maybe Davisson-Germer should be over ;-)

So, in the new paper mentioned on Peter Woit's blog, (Realization of a gravity-resonance-spectroscopy technique, Tobais Jenke, et. al., Nature Physics 17 April 2011), what is the resolution of the spectroscopy? What kinds of deviation from a linear gravitational potential, g*x, can it be used to measure?

Is the Schroedinger equation considered consistent with general relativity? In other words, if this experiment were conducted near a black hole where the gravitational field varied significantly over the physical area of the experiment, could you still use Shroedinger's equation to solve it using some variable expression in place of the constant g?

The Schrödinger equation isn't consistent with General Relativity because it's non-relativistic. You can take the potential for a particle in the Schwarzschild-metric and put it in the Schrödinger equation, that will do as long as the particle is moving slowly.

"The Schrödinger equation isn't consistent with General Relativity because it's non-relativistic."

This is an urban legend, isn't it?

The Schrödinger equation is a legitimate relativistic equation for spin-0 fields (like Klein-Gordon) but you have to quantize in the Schrödinger picture for the equation to make sense.

As usual the field is seen as a collection of harmonic oscillators and in their Schrödinger equation you just replace their coordinates with fields, the wave function with a wave functional etc. This formulation for practical reasons is rarely used in QFT though.

The wrong impression that the Schrödinger equation is non-relativistic is due to the way QFT was developed historically (i.e. because fields were seen as wave functions initially) and that's why the term 'second quantization' is used until today.

"The Schrödinger equation isn't consistent with General Relativity because it's non-relativistic."

This is an urban legend, isn't it?

The Schrödinger equation is a legitimate relativistic equation for spin-0 fields (like Klein-Gordon) but you have to quantize in the Schrödinger picture for the equation to make sense.

As usual the field is seen as a collection of harmonic oscillators and in their Schrödinger equation you just replace their coordinates with fields, the wave function with a wave functional etc. This formulation for practical reasons is rarely used in QFT though.

The wrong impression that the Schrödinger equation is non-relativistic is due to the way QFT was developed historically (i.e. because fields were seen as wave functions initially) and that's why the term 'second quantization' is used until today.

I don't want to quantize anything. I was answering a question. The question was whether you can use the Schrödinger equation if you don't have Newtonian gravity, but General Relativity. My answer is, in general no, because the Schrödinger equation isn't Lorentz-invariant to begin with, so you can't even use it for Special Relativity. As long as you're particles are non-relativistic tough, you should be fine if you replace the potential.

"because the Schrödinger equation isn't Lorentz-invariant to begin with, so you can't even use it for Special Relativity"

This is because when people hear about the Schrödinger equation they immediately think (due to the way many text books present the subject for historical reasons) of the old quantum mechanics (before QFT) where particles are described by wave-functions. This is not correct though because Schrödinger equation is a general equation which can be used in a relativitistic QFT formulation but then you have to use the Schrödinger picture which is rarely used; this doesn't mean though that the Schrödinger equation (as functional differential equation which uses instead of a wave function a wave functonal of the field and with the appropriate Hamiltonian which is the key) is non-relativistic and thus not compatible with special relativity. There is nothing wrong with it and is valid in all cases.

Within GR where there is no time evolution in the strict sense it takes the form of Wheeler-De Witt equation.

Giotis: I think we're talking past each other. I am talking about the equation that has been used in the above post, which I assume is what we are talking about, and the one I thought the above question was referring to. The equation is i \partial_t \psi = - (1/2m \nabla^2 + V) \psi. It is, as I said several times now, not Lorentz-invariant.

Ok I see, I didn't realize that you were referring to the specific equation in the post. I thought you were talking about the Schrödinger equation in general; but then again why you said you have no idea of what I'm talking about? You could have made this clarification in your first reply. Anyway it's not important...